An Airpets employee at the company's site near Heathrow's Terminal 5. Photograph: Sean Smith for the Guardian

"We'll take anything," says Angela Lacey, a pet importer working in the shadow of Heathrow's Terminal 5.

Hundreds of thousands of dogs, cats, guinea pigs, tropical fish, parrots, rabbits and even the odd rat have passed through the Airpets handling and quarantine centre since its founder converted it from a pig farm 47 years ago.

"But we try to put [customers] off hamsters," says Lacey. Her colleagues break out into giggles. "They don't make it," she adds. "They don't live very long, little hamsters. The flight coming over, then the [four-month] quarantine. The average lifespan is, what, a couple of years? I wouldn't pay all that money for a hamster, no I don't think so."

And it is a lot of money. Flying in a hamster from New York plus the legally enforced four-month stay in the quarantine cages would cost at least £2,000.

The cost of sending a pet overseas depends on the animal's size, but Nick Foden-Ellis, Airpets managing director, says it is "rarely less than a business-class ticket for the same route". A recent job sending a chocolate labrador to Sydney cost more than £4,000.

Despite the economic downturn business has been booming for Airpets and the dozens of other pet importing and exporting firms. "We've had the busiest summer for 10 years," says Foden-Ellis.

The firm has recorded a recent rise in pet exports to Australia after its currency fell against the pound, he says. The pound has recovered to A$1.71 from A$1.44 in March. "The exchange rate change adds a huge amount to disposable income. People that might not have been able to afford to take their pets with them and might have left them behind with a relative now feel they can afford to take them with them."

He says exports to Australia – the most expensive destination as it requires a stopover – are up 8%-10%. Australia is Airpets' biggest export destination accounting for 40% of all pet travel. "Everyone has been watching [ITV show] Poms in Paradise," he says.

Official figures from the Animal Health and Veterinary Laboratories Agency show 139,000 dogs were brought into the UK last year compared with 85,000 in 2011. Imports of cats jumped 74% to more than 14,000. Even ferrets, popularised in recent years by Paris Hilton and Jonathan Ross rose from 68 in 2011 to 93 in 2012.

The agency, part of the Department for Environment, Food and Rural Affairs, also reports a big rise in families importing rabbits, chinchillas, rats, guinea pigs, fish, snakes and even frogs.

It says imports of cats and dogs to the UK exploded on 1 January last year when the pet passport scheme made it much easier for people to move pets from the European Union and certain other countries to the UK.

"Overnight, it meant the end of mandatory quarantine for thousands of animals that had rabies jabs and vet certificates," says Foden-Ellis. "We opened at midnight on New Year's Day as some of our animals were eligible to leave without seeing out the rest of their sentence. Four or five people came in to collect their pets, we had some champagne."

While it is now much easier to bring in animals, it still requires planning with vaccinations for rabies and check-ups several weeks in advance.

"People don't think about how tricky it can be," says Foden-Ellis, a former paratrooper. "An awful lot of our customers are people who go on holiday to north Africa, Cyprus or Spain and fall in love with a street dog and want to bring it home with them. They'll manage to get it on a flight back from Spain or wherever, but when they arrive here they'll be asked for documentation. 'Well, I haven't got it,' they'll say. That means it will have to go into quarantine, which can cost £1,500-£2,000."

Sadly, he says, when some people find out the size of the bill they fall out of love. "We will never destroy an animal. That's why if you check out these guys' staff profiles you'll see how many pets they've got. They're predominately strays that people haven't been able to afford the quarantine bill for."

While the number of strays Airpets handles is rising, they are a tiny proportion compared with the pets which executives take when posted abroad. "There has been a huge rise in pets going out to Asia and Bric [Brazil, Russia, India, China] countries," he says. "As those economies grow businesses are sending their people there, and animals are becoming more accepted as family pets in Asia."

Laura Nolan, an Airpets customer who took her cat, Gizmo, from Plymouth to Sydney, says the trip cost up to £16,000 including her own flight and visas. "We had her since she was a kitten and we both agreed that we wouldn't have been able to leave her behind and not know what happened to her." Despite the cost, Nolan adds, she will take Gizmo with her if she ever has to move again.

Foden-Ellis also boasts a string of celebrity customers, some of whom even fly out their pets with them on holiday. "We have movie stars, footballers, everyone," he says. One of the world's best-known footballers regularly books his dogs on transatlantic flights, he lets slip, before threatening to hunt me down if the Guardian names him.

Mike Sawyer, sales and development manager of rival pet transporters JCS Livestock, says his firm has also recorded an increase in business. "We have got a couple of people who take their pets on holiday with them; one customer takes his dogs to New York back and forth at least twice a year and another client sends his dogs to Australia and back," he says. "Another client shipped their dog to America for a two-week holiday. Luckily it was a small dog – you pay by the size."

Sawyer points out that the airlines make much more than the handlers. Ramón Delima, vice-president of special cargo at Air France-KLM, which claims to transport more animals than any other airline, says animal transport does not follow socioeconomic trends because pets are largely seen "as part of the family".

He refuses to say how much money KLM makes from transporting animals, but it's a much higher margin than regular cargo. "It's not as simple as transporting a box. We have an animal hotel in Schiphol [airport near Amsterdam]. We take full care of them … dogs are walked in the park, literally. We have animal attendants, just like cabin attendants for passengers. This is not just putting an animal in a box and sending it off."